Solve for $x$ and $y$ using elimination. $\begin{align*}4x+4y &= 7 \\ x-6y &= -7\end{align*}$
We can eliminate $y$ when its corresponding coefficients are negative inverses. Recalling our knowledge of least common multiples, multiply the top equation by $3$ and the bottom equation by $2$ $\begin{align*}12x+12y &= 21\\ 2x-12y &= -14\end{align*}$ Add the top and bottom equations. $14x = 7$ Divide both sides by $14$ and reduce as necessary. $x = \dfrac{1}{2}$ Substitute $\dfrac{1}{2}$ for $x$ in the top equation. $4( \dfrac{1}{2})+4y = 7$ $2+4y = 7$ $4y = 5$ $y = \dfrac{5}{4}$ The solution is $\enspace x = \dfrac{1}{2}, \enspace y = \dfrac{5}{4}$.